On a high photocatalytic activity of high-noble alloys Au–Ag/TiO2 catalysts during oxygen evolution reaction of water oxidation

The analysis via density functional theory was employed to understand high photocatalytic activity found on the Au–Ag high-noble alloys catalysts supported on rutile TiO2 during the oxygen evolution of water oxidation reaction (OER). It was indicated that the most thermodynamically stable location of the Au–Ag bimetal-support interface is the bridging row oxygen vacancy site. On the active region of the Au–Ag catalyst, the Au site is the most active for OER catalyzing the reaction with an overpotential of 0.60 V. Whereas the photocatalytic activity of other active sites follows the trend of Au > Ag > Ti. This finding evident from the projected density of states revealed the formation of the trap state that reduces the band gap of the catalyst promoting activity. In addition, the Bader charge analysis revealed the electron relocation from Ag to Au to be the reason behind the activity of the bimetallic that exceeds its monometallic counterparts.

In the current era, solar energy is confined for practical reasons and has increased the attention of both academia and industrial sectors. The essential worldwide goal of solar energy is to create electricity, and it is prospective to generate fuels from the waste of CO 2 and H 2 O by induction of solar photochemistry. In the past 40 years, heterogeneous photocatalysis and photochemistry fields have developed comprehensively in return to demanding energy and environmental issues [1][2][3][4][5][6][7][8] . One of the most capable solutions for the energy problem is photocatalytic overall water oxidation (POWS), i.e., oxygen and hydrogen evolution 9,10 . Water electrolysis-one of the most practical routes to produce hydrogen to establish clean and renewable energy cycles, accounted for up to 4% of the global hydrogen production 11,12 . Water electrolysis entails two half-cell reactions: the cathodic hydrogen evolution (HER) and anodic oxygen evolution (OER) reactions.
The metal/semiconductor photocatalysts have vast POWS potential and have drawn broad attention for a long time 13 . Since 1972, a typical semiconductor TiO 2 exhibited a suitable band structure for POWS as a photocatalyst 14 . Moreover, the TiO 2 -supported Pt catalysts are competent of POWS under the range of ultraviolet light 15 . Evidently, the visible light POWS shows a higher potential for practical applications. However, due to the large bandgap of TiO 2 , TiO 2 -based photocatalysts POWS driven by visible light remains a great challenge [16][17][18] . Even though the bandgap of TiO 2 could be effectively narrowed down by elemental doping to harvest visible light, the energy levels introduced by impurity dopants can act as recombination centers for electrons and holes, resulting in a poor photocatalytic activity 19,20 .
Moreover, it has been reported that various metals, e.g., Pt, Au, Ag, Pd, Ru, Cu, etc., loaded on semiconductors materials, e.g., TiO 2 , RuO 2 , etc. to confine photoexcited electrons through the formed Schottky junction can result in reserved charge recombination for photocatalytic enhancement [21][22][23] . When the material is operated in a visible light range, Au, Ag, and Cu can act as metallic plasmons and trigger localized surface plasmon resonance (LSPR) 24,25 . This energizes the electrons above the Fermi level from the occupied energy levels 26,27 . These electrons with high energy pass through the Schottky barrier of metal-semiconductor can be located in the semiconductor  www.nature.com/scientificreports/ Au (5d). The Ag-5 s orbital is much higher in energy than the Au-6 s orbital energy, so a partial charge transfer from Au to Ag occurs 48 .
Consequently, Au atoms tend to be negatively charged, while Ag atoms tend to be positively charged. The partial charge transfer from Au atoms to Ag atoms gives remarkable electrostatic stabilization, making the alloy formation more favorable than pure gold and silver clusters. Therefore, the equivalent mixing between Au and Ag atoms in the alloy formation will likely be more preferential. For Au 4 /TiO 2 and Ag 4 /TiO 2 tetrahedral structures positioned on bridging oxygen rows, vacancy results are shown in S.I (see fig S10 and S11).
It is well known that gold clusters have the lowest spin multiplicity as the ground state 49 . Previously, the spin effects of Au clusters on TiO 2 (110) 50 and other metal oxides 51 were studied. As a result, spin and electron distributions were assumed to have no effects in the case of noble metal atoms. Figure 1 shows that the spin-up and spin-down states of the DOS profiles of Ag 4d and Au 5d in Ag 4 Au 4 /TiO 2 are symmetrical, while in the case of Au 4 /TiO 2 , it exhibits slightly asymmetrical DOS shown in Figure S10 of the supplementary document.
TiO 2 surface can donate electrons and is ascribed to excess electrons from the defect state found inside the bandgap 34,52-54 . Ag and Au, relatively electronegative atoms, prefer to take the missing oxygen when creating an oxygen vacancy. Pulling an oxygen atom out of the bridging oxygen atom releases electron density, creating a vacancy site. This excess electron is transferred to Ag and Au atoms binding to the vacancy site. This implies that the support plays an active role in catalytic activity because it alters its chemical properties by transferring charge to Ag and Au 55 . In this work, DFT analyses are employed to understand the water oxidation mechanism on the interface between Au/Ag high noble alloys catalysts supported on rutile TiO 2 (110) and such a mechanism on the support itself. Thus, two main issues are focused, (1) determination of active regions on the surface (2) thermodynamically stable size and location of the active site cluster of initial (Au/TiO 2 , Ag/TiO 2 , Au-Ag/TiO 2 ) and cluster models (Au 4 /TiO 2 , Ag 4 /TiO 2 , Au 2 -Ag 2 /TiO 2 ).
Catalytic performance of TiO 2 support during oxygen evolution reaction (OER). We first discuss the water oxidation mechanism on clean rutile (110) TiO 2 surface. In Fig. 7d, we showed the pure surface of TiO 2 rutile (110), which is well investigated by many researchers theoretically and experimentally. The reported overpotential value for the pure surface is 0.80 eV with the rate-determining step OH* as shown with recreation in Fig. 2 33,56 . Furthermore, Malik et al. and Norskov et al. reported that on the clean or pure surface of TiO 2 mechanism of water oxidation proceeds through surface-bound peroxo O* species. Thus, Ti was the only metal active site choice for studying water oxidation mechanisms.
Determination of active regions on the catalyst. Now we started analyzing with initial models of Au/TiO 2 , Ag/TiO 2 , and Au-Ag/TiO 2 , where the model shown in Fig. 1a comprises one Ag atom replacing the bridging oxygen atom on the TiO 2 surface, while in Fig. 7b, one Au atom substituted an oxygen atom on the TiO 2 surface. When we studied Au-Ag/TiO 2 model (Fig. 7c), we tried different positioning of Au and Ag atoms, such as placing both atoms next to each other in the same bridging row or placing them on the same bridging row with the gap of one bridging oxygen atom in between Au and Ag (In S.I. see figure S1). The most stable position was when Au and Ag atoms were positioned on two bridging oxygen vacant sites in different rows, as shown in Fig. 7c. To study the OER mechanism on the Au-Ag/TiO 2 slab structure, the Ti atom was first chosen as an active site, and then the calculations were performed by choosing Au and Ag as an active site. For Au/TiO 2 and Ag/TiO 2 slab structures, the free energy diagrams are shown in S.I. in Figure S8 (a) and (b). The Ti active site was selected far from noble metal atoms. The OER data of the initial model for Fig. 7c, i.e., Au-Ag/TiO 2 , is listed in Table 1. The OER data for Fig. 7a,b are shown in S.I. in table S2, and intermediates are shown in Figure S3 and S4.
From the OER results shown above in Fig. 3, it is indicated that the Au atom is preferably the active site with the lowest overpotential with an initial model. However, since Au and Ag atoms are placed on different bridging rows of oxygen vacancy sites, there is no interaction between noble metal atoms on the TiO 2 surface. www.nature.com/scientificreports/ OER performance of TiO 2 supported Au-Ag high-noble alloys catalysts. An initial model is good enough to determine the active site; yet, Au and Ag atoms were placed on different bridging rows, so there was no interaction between Au and Ag atoms. Hence, no charge transfer could take place from Ag to Au. To understand the water oxidation mechanism at the interface of noble metal atoms cluster and TiO 2 metal oxide surface, we further extended calculations to use a 3-dimension tetrahedral cluster of (Au and Ag) atoms on the TiO 2 surface because of its stability. This structure we refer to as a cluster model. Our previous work 34 showed that the Au 2 -Ag 2 /TiO 2 structure is more suitable when embedded in the location of oxygen vacancy. Nevertheless, we show how the OER activity is affected when the tetrahedral cluster is placed on a stoichiometric surface at the two bridging oxygen rows (a different position), as shown in Fig. 8c. In the supplementary document, we have demonstrated OER intermediates of Au 4 /TiO 2 , Ag 4 /TiO 2 , and Au 2 -Ag 2 /TiO 2 when placed on two bridging rows (see figure S5 and S6). The comparative free energy diagram for Au 4 /TiO 2 , Ag 4 /TiO 2 on two bridging rows is given in figure S9 (a) and (b) with the OER data (relative free energies) shown in table S3. Figure 4a shows the results for Au 2 -Ag 2 /TiO 2 on two different bridging rows. The comparative free energy profile shows that the lowest overpotential is when Au was chosen as an active site. However, the OH*, O*, and OOH* steps were strongly bonded for the Ti active site than Au and Ag. Similar findings were observed for the model when Au 2 -Ag 2 /TiO 2 was placed on a bridging row oxygen vacancy. The O 2 evolution was preferred on the Au site, free energy diagram shown in Fig. 4b. From the above-shown results, it has been seen that either Au 2 -Ag 2 cluster adsorb on TiO 2 surface by connecting to two bridging oxygen rows or an oxygen vacancy is created, and then the cluster is anchored into the vacant site; the perfect active site is Au. In all cases, Ti is the least preferred due to high overpotential. It was also found that the rate-determining step for OER on Au 2 -Ag 2 / TiO 2 surface is the O* intermediate, i.e., 2nd step when the active site is noble metal atoms, i.e., Au or Ag. While the rate-determining step for OER with Ti as an active site is OOH* intermediate, i.e., the 3rd step. For Au 2 -Ag 2 / TiO 2 , the perfect active site is Au. When applied bias (U = 0 V), all steps go uphill thermodynamically.
When Au 2 -Ag 2 /TiO 2 cluster is adsorbed on two bridging rows, TiO 2 surface is stoichiometric; however, a remarkable change is noticed in overpotential when embedded on a vacant oxygen site. This change is attributed to the presence of oxygen vacancy because Au and Ag atoms are higher in electronegativity and prefer to stay in the place of oxygen vacancy. Therefore, there is excess electron density when the oxygen atom from the bridging row is removed. This "excess electron density" attaches the silver and gold clusters (with the ability of electronegativity) to the vacancy site. Moreover, when gold and silver atoms are bonded together in bimetallic clusters, there is a charge transfer from Ag to Au, which is accountable for making Au negatively charged and expected to be more active. Table 1. The relative free energies for each type of active site on the bimetallic Au-Ag/TiO 2 system shown in Fig. 7c, starting from the dissociation of a water molecule (H 2 O + * ↔ H* + HO*) set as the reference point with zero energy. The propagating step is denoted as ΔG OH* , ΔG O* , and ΔG OOH* shown together with their overpotentials and rate-determining step. *The rate-determining steps. www.nature.com/scientificreports/ For the cluster systems, the relative free energies of all the intermediates (OOH*, OH*, and O*) relative to the starting state (* + 2H 2 O) are shown in Table 2. For all systems, we see that the trend in activity is the same for the active site, i.e., Au > Ag > Ti. For noble metal atoms, the rate-determining step is the O* step, and for transition metal, the rate-determining step is OOH*. Notably, the OOH* step is much less stable when the active site is Ti (blue curves in free energy diagrams) conversely the OH* and O* are more stable and vice versa for the noble metal active sites case (red and green curve for Au and Ag respectively in free energy diagrams).  www.nature.com/scientificreports/ OER performance of TiO 2 supported Ag and Au catalysts. As explained earlier, adsorption of Ag on TiO 2 surface becomes more stable when interacted with Au. Moreover, the photocatalytic property of Ag species is enhanced because Au promotes charge transfer from Ag to Au on the TiO 2 surface. To explain this phenomenon in-depth, we aimed to study the effect of only Au on TiO 2 and Ag on TiO 2 with cluster and initial model. We used the same strategy for choosing the active site and stable structure mentioned earlier for the bimetallic cluster. However, we only discuss the results of Au4/TiO 2 and Ag4/TiO 2 cluster models shown in Fig. 10a,b, and Table 3, respectively. Free energy diagrams are shown in Fig. 5a,b and for OER intermediates (see Table 3. The relative free energies for each type of active sites on the monometallic Au 4 /TiO 2 and Ag 4 /TiO 2 cluster bridging row oxygen vacant site shown in Fig. 7a,b,  www.nature.com/scientificreports/ S.I. figure S7). For Au 4 /TiO 2 and Ag 4 /TiO 2 clusters placed on two bridging rows, free energy diagrams are given in S.I figure S9. Studying the OER mechanism on all Au and Ag clusters, i.e., Ag 4 /TiO 2 and (b) Au 4 /TiO 2 , placed in bridging row oxygen vacancy, we found that overpotential for Ag active site was 0.53 V higher for Ag 4 /TiO 2 when compared with Ag 2 -Au 2 /TiO 2 , i.e., 0.70 V. However, for the Ti active site, overpotential is remarkably less, and a difference of 0.81 V was observed. If we compare Au 4 /TiO 2 with Ag 2 -Au 2 /TiO 2 with Au active site, the overpotential remains the same almost. However, there are noticeable changes for the Ti active site, i.e., 1.39 V. So we conclude from our results that, in the case of any model, either initial or cluster, the active site is the preferably Au atom. In terms of overpotential, the trend of photocatalytic activity on all studied models is Au > Ag > Ti.

Insights into OER.
We mentioned earlier that Ag noble metal could be an excellent approach over Au for plasmonic photocatalytic activity; however, the plasmonic frequency of Ag is near the UV region responsible for limiting the catalytic activity of Ag-based photocatalyst. Therefore, we make a bimetallic alloy of Au and Ag. From our OER calculations, we have concluded that on Ag 4 /TiO 2 over potential is higher for Ag active site, i.e., 1.23 V, for Au 4 /TiO 2 overpotential is lower with Au active site, i.e., 0.58 V. But when Au and Ag are in a bimetallic cluster state, and we perform OER calculations on Au 2 -Ag 2 /TiO 2 , the overpotential with Au and Ag active site is 0.60 V and 0.70 V, respectively. Hence, this proves that the presence of Au noble metal could stabilize the Ag and lead to better photocatalytic activity. Unfortunately, there are fewer reports of OER studies on noble metalloaded TiO 2 structures. Wang et al. 57 reported the OER mechanism on Au/TiO 2 initial structure. Instead of removing Bridging oxygen, surface oxygen was removed and was further doped with Au atom. They only chose Ti as an active site. The rate-determining step was OOH* with an overpotential of 1.77 V. To provide further insight about noble metal-doped TiO 2 surface, the calculations were extended by using a small (four atoms) tetrahedral cluster for determining the OER activities. In the near future, studies with larger bimetallic clusters are needed to check if the trends observed here stay within the range of bimetallic particles.

Bader charge analysis.
Oxygen vacancy is one of the vital defects among all the defects known in TiO 2 58 . It has been discovered that oxygen vacancies can perform as significant adsorption and active sites for heterogeneous catalysis, which can sturdily sway the reactivity of metal oxides 59 . Theoretically, oxygen vacancies formation on TiO 2 is direct to the creation of unpaired electrons or Ti 3+ centers, which possibly will form donor levels in the electronic structure of TiO 2 60 . Theoretical and experimental results indicate that the excess electrons are located on the oxygen vacancy states. When noble metal atoms (Au or Ag) are adsorbed on the TiO 2 surface, they tend to adsorb on the oxygen vacant site due to the high electronegativity of noble metal atoms.
We performed the Bader charge analysis to investigate the electron transfer between the metal active site (Au, Ag, or Ag-Au) and the catalyst surface for Ag 4 /TiO 2 , Au 4 /TiO 2 , and Ag 2 -Au 2 /TiO 2 system, as shown in Fig. 6. The calculated values are given in Table 4. The negative and positive signs refer to electron accumulation and depletion, respectively.
For the Ag 4 /TiO 2 structure, the positive value of the Ag 4 cluster (+ 0.14 |e|) designates its role as an electron donor during its adsorption on the TiO 2 surface with an oxygen vacancy site. Moreover, a negative Bader charge of Ag1 (− 0.23 |e|) together with the positive Bader charge of other surrounding Ag atoms (varies from + 0.04 to + 0.23 |e|) reveal electron transfer from the oxygen vacancy site to other Ag atoms: Ag2, Ag3, and Ag4, to Ag1 atom. Hence, the activity of the Ag1 atom is improved.
For Au 4 /TiO 2 structure, the Au 4 cluster plays a role as an electron acceptor, taking electrons from the TiO 2 surface. Similar to the Ag 4 /TiO 2 structure, the most negative Bader charge is found on the Au1 site, which specifies that the activity of the Au1 site is enhanced after its adsorption on the TiO 2 surface with oxygen vacancy site. In addition, the electron accumulations around the Au1 site is found to be two times higher than that of the Ag1. This suggested a high activity of the Au system.
For the Ag 2 -Au 2 /TiO 2 system, the whole bimetallic Ag 2 -Au 2 cluster acted as an electron acceptor similar to the Ag cluster in the Ag 4 /TiO 2 system. However, the total Bader charge value of the Ag 2 -Au 2 cluster is less www.nature.com/scientificreports/   www.nature.com/scientificreports/ negative (− 0.14 |e|). Considering the two-atom Ag (Ag 2 ) and two-atom Au (Au 2 ) cluster within the Ag 2 -Au 2 cluster, it was found that the Au 2 subcluster holds a negative Bader charge of − 0.69|e|, while the positive Bader charge was found in the Ag 2 subcluster of + 0.55 |e|. This phenomenon describes the additional electron transfer from the Ag 2 to the Au 2 subcluster. In addition, the activity of the Au1 active site in the Ag 2 -Au 2 cluster is promoted via the electron transfer from the Ag site. Moreover, the Au1 site located at the vacancy site of the TiO 2 surface still has the most negative Bader charge value of − 0.48 |e|. This Bader charge value of the Au1 atom in the Ag 2 -Au 2 cluster is similar to the Au1 atom in the Au 4 cluster. Hence, the presence of Ag species in the Ag 2 -Au 2 bimetallic cluster facilitates electron delocalization around Au1 and Au2, creating another active site-the Au2 site. Therefore, the activity of the bimetallic Ag 2 -Au 2 cluster exceeds the monometallic cluster of Au4 and Ag4 clusters due to the presence of the Ag 2 subcluster, which donates additional electrons to the Au 2 subcluster.

Conclusion
In summary, we used density functional theory to investigate the role of Au metal in Au-Ag high noble alloys catalysts supported on TiO 2 on the performance during the oxygen evolution of water oxidation (OER), in which the catalysts are modeled as Au, Ag, and Au-Ag supported on rutile TiO 2 (110). Since doping of noble metal atoms increases the photocatalytic activity of TiO 2 , the Ag noble metal is shown to have a promising catalytic activity among the groups. However, studying the reaction mechanism on such a surface is essential due to its low stability.
Combining Au (a more stable noble atom) into Ag-based TiO 2 forming bimetallic high-noble alloy catalysts can enhance photocatalytic activity. The study investigated the high-noble alloy cluster in terms of the most stable configuration verified by the most stable position of the metal-support interface, stability of the active site, optimal size, and the most active region of the catalyst for the OER during water oxidation. The most stable location of the metal-support interface for the Au-Ag high-noble alloy catalyst modeled by the Au 2 -Ag 2 /TiO 2 was found at the bridging row oxygen vacant site. On the performance of this high-noble alloy catalyst, the Au atom is always preferred as an active site for OER regardless of the size or position of the cluster. The photocatalytic activity trend indicated by an overpotential and active site preference is Au > Ag > Ti. Hence, in both cases of monometallic (Au and Ag) and bimetallic (Au-Ag), the Ti active site is least preferred.
With this positioning, the overpotential is much lower for the Au and Ag active site, i.e., 0.60 V and 0.70 V, indicating that the Au atom stabilizes Ag. Therefore, utilizing the high-noble alloy catalyst of Au-Ag can improve the oxygen evolution activity on the rutile TiO 2 (110) surface during the water oxidation reaction, promoting efficient hydrogen production and supporting the hydrogen economy.
In addition, the future design of high photocatalytic performance catalyst based on this study must consider the profiles from the Bader charge analysis, which suggested that the presence of the Ag atom can stabilize the Au atom via the electron transfer to the Au, where this generated the trap state between the valence band maximum (VBM) and the conduction band minimum (CBM) reducing the band gap of the catalyst promoting activity as supported by the projected density of states (DOS) profile of this Au-Ag/TiO 2 catalyst.

Methods
Computational details. Density functional theory-based calculations. We employed the first-principles spin-polarized density functional theory (DFT) calculations using the projector augmented wave (PAW) method applied in the Vienna Ab initio simulation package (VASP) 61,62 . For the exchange-correlation functional, the generalized gradient approximation (GGA) 63 of Perdew − Burke − Ernzerhof (PBE) 64 was used with the planewave cutoff energy of 400 eV. Ultra-soft pseudo-potential, the interaction between valence electrons and the ionic core 65 , is used with 2s 2 2p 4 , 3d 2 4s 2 ,4d 10 5s 1 , 5d 10 6s 1 as the valence electrons configuration for the O, Ti, Ag, and Au atoms, respectively. The Monkhorst and Pack scheme of k-point sampling was employed to integrate the first Brillouin zone 66 , and a 3 × 2 × 1 grid was used to obtain the geometry optimization and total energies for the rutile TiO 2 (110) surface. The applied residual forces for geometry optimization and convergence on the atoms were 0.01 eV/Å, and the self-consistent iteration awaits the tolerance for total energy to get to 10-4 eV. The optimized lattice parameters a = 11.83 Å, b = 12.99 Å, c = 30.08 Å, and α = β = γ = 90.00° for rutile (110). The calculations were done on a four-layered rutile TiO 2 (110) surface slab with a super-cell of 3 × 2 × 1 where the bottom layer was fixed at bulk-truncated positions and the others were fully relaxed. Dipole correction was applied to all of the calculations. We calculated the energy position, valence band maximum (VBM), conduction band minimum (CBM), and trap states level for band structure calculations. For these properties, we carried out single-point energy calculations with the HSE06 functional 67 because they are known to provide a better description of the valence band (VB) and conduction band (CB), band edges, and band gaps (that are underestimated with PBE).
Catalyst model constructions. The OER mechanism was studied on the Au/TiO 2 , Ag/TiO 2 , and Au-Ag/ TiO 2 . For the initial model: We created an oxygen vacancy in the bridging row on the surface which was filled by gold atom in Au/TiO 2 model (Fig. 7a) and similarly Ag atom for Ag/TiO 2 (Fig. 7b); however, in the case of the Au-Ag/TiO 2 model, we replaced two bridging oxygen atoms from different bridging rows on the surface to replace Au and Ag atoms shown in Fig. 7c. Figure 7d shows the clean rutile (110) surface. We have demonstrated the bridging oxygen, which was pulled out to create oxygen vacancies. These configurations are shown to be at the most stable metal-support interface location. For the cluster model: We placed the tetrahedral cluster of 4 atoms Au 4 , Ag 4 , and Au 2 -Ag 2 with different orientations on the surface, as shown in Figs. 8 and 9. We chose the cluster as a model because of its stability 68  Evaluation of water oxidation performance. The reaction mechanism of water oxidation can occur with four proton-coupled electron transfer steps, where the initial step is the water adsorption process. Our calculations showed that water molecules tend to adsorb on the TiO 2 surface. We used the mechanism proposed by Rossmeisl et al. 69 . The OER mechanism is progressed in the following four steps as shown in equations 1-4, where ½ H 2 ↔ H + + e − is the half electrode reaction.   www.nature.com/scientificreports/ The OER intermediates are shown in Fig. 10a-f on Ti and Au as an active site, for Ag active site (see figure S2) in S.I. To evaluate the steps as mentioned above, it is expedient to characterize the free energies ΔG OH* , ΔG O* and ΔG OOH* of the OH*, O*, OOH* intermediate states, and ΔG O2 for the O 2 final stage, all determined relative to the resting state (* + 2H 2 O). In Sect. 1 of supplementary information (S.I), we have details of the determination of the free energies (including vibrational and entropy corrections). We can extract the reaction step-free energies from the relative free energies; ΔG n (n = I, II, III, and IV) are written as equations 5-6.
The overpotential of reactions could be associated reliably to the proton and electron transfer to adsorbed species strongly bonded to the surface at the electrode potential. This reaction series starts with (* + 2H 2 O) on a metal active site leading to OH*, shown in steps I-IV. The diffusion of species and other surface reactions most likely depends weakly on the potential.
The theoretical overpotential is defined as:

Data availability
All data generated or analyzed during this study are included in this published article (and its Supplementary Information files).